Pub Date : 2015-04-30DOI: 10.3846/13926292.2016.1138418
J. Zhao, E. Vollebregt, C. Oosterlee
In this paper we extend the range of applicability of the boundary element method (BEM) for concentrated elastic contact problems by computing the influence coefficients (ICs) numerically. These ICs represent the Green's function of the problem, i.e. the surface deformation due to unit loads. For the half-space they are analytically available. An elastic model is proposed to compute these coefficients numerically, by the finite element method (FEM). We recommend proper strategies of FEM meshing and element types, considering accuracy and computational cost. The effects of computed ICs to contact solutions are examined for a Cattaneo shift problem. The work in this paper aims at providing a reference to study fast solvers for the conformal contact problem where ICs are not analytically known.
{"title":"Extending the BEM for elastic contact problems beyond the half-space approach","authors":"J. Zhao, E. Vollebregt, C. Oosterlee","doi":"10.3846/13926292.2016.1138418","DOIUrl":"https://doi.org/10.3846/13926292.2016.1138418","url":null,"abstract":"In this paper we extend the range of applicability of the boundary element method (BEM) for concentrated elastic contact problems by computing the influence coefficients (ICs) numerically. These ICs represent the Green's function of the problem, i.e. the surface deformation due to unit loads. For the half-space they are analytically available. An elastic model is proposed to compute these coefficients numerically, by the finite element method (FEM). We recommend proper strategies of FEM meshing and element types, considering accuracy and computational cost. The effects of computed ICs to contact solutions are examined for a Cattaneo shift problem. The work in this paper aims at providing a reference to study fast solvers for the conformal contact problem where ICs are not analytically known.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125898594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-12-31DOI: 10.4208/NMTMA.2016.M1422
Xin He, C. Vuik
In this report we explore the performance of the SIMPLER , augmented Lagrangian, ’grad-div’ preconditioners and their variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem Reynolds number varies from a low to the limiting number for a laminar flow.
{"title":"Comparison of some preconditioners for the incompressible Navier-Stokes equations","authors":"Xin He, C. Vuik","doi":"10.4208/NMTMA.2016.M1422","DOIUrl":"https://doi.org/10.4208/NMTMA.2016.M1422","url":null,"abstract":"In this report we explore the performance of the SIMPLER , augmented Lagrangian, ’grad-div’ preconditioners and their variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem Reynolds number varies from a low to the limiting number for a laminar flow.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134120841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with fast and reliable numerical solution methods for the incompressible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial differential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equations. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that effects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.
{"title":"On preconditioning incompressible non-Newtonian flow problems","authors":"Xin He, M. Neytcheva, C. Vuik","doi":"10.4208/jcm.1407-m4486","DOIUrl":"https://doi.org/10.4208/jcm.1407-m4486","url":null,"abstract":"This paper deals with fast and reliable numerical solution methods for the incompressible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial differential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equations. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that effects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124147597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-08-31DOI: 10.3997/2214-4609.20143232
P. Slingerland, C. Vuik
Accurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations, we have cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p=0. This paper provides a numerical comparison of the performance of both two-level methods in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice for the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to faster and more accurate fluid pressure simulations.
{"title":"Fast linear solver for pressure computation in layered domains","authors":"P. Slingerland, C. Vuik","doi":"10.3997/2214-4609.20143232","DOIUrl":"https://doi.org/10.3997/2214-4609.20143232","url":null,"abstract":"Accurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations, we have cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p=0. This paper provides a numerical comparison of the performance of both two-level methods in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice for the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to faster and more accurate fluid pressure simulations.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2012-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116215029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give two important generalizations of the Induced Dimension Reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi-shift Quasi-Minimal Residual IDR (QMRIDR) variant. Numerical examples are presented to show the effectiveness of these new IDR variants compared to existing ones and to other Krylov subspace methods
{"title":"Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems","authors":"M. Gijzen, G. Sleijpen, J. Zemke","doi":"10.15480/882.1019","DOIUrl":"https://doi.org/10.15480/882.1019","url":null,"abstract":"We give two important generalizations of the Induced Dimension Reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi-shift Quasi-Minimal Residual IDR (QMRIDR) variant. Numerical examples are presented to show the effectiveness of these new IDR variants compared to existing ones and to other Krylov subspace methods","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126435095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Swing options give contract holders the right to modify amounts of future delivery of certain commodities, such as electricity or gas. In this paper, we assume that these options can be exercised at any time before the end of the contract, and more than once. However, a recovery time between any two consecutive exercise dates is incorporated as a constraint to avoid continuous exercise. We introduce an efficient way of pricing these swing options, based on the Fourier cosine expansion method, which is especially suitable when the underlying is modeled by a Levy process.
{"title":"An efficient pricing algorithm for swing options based on fourier cosine expansions","authors":"Baocheng Zhang, C. Oosterlee","doi":"10.21314/JCF.2013.268","DOIUrl":"https://doi.org/10.21314/JCF.2013.268","url":null,"abstract":"Swing options give contract holders the right to modify amounts of future delivery of certain commodities, such as electricity or gas. In this paper, we assume that these options can be exercised at any time before the end of the contract, and more than once. However, a recovery time between any two consecutive exercise dates is incorporated as a constraint to avoid continuous exercise. We introduce an efficient way of pricing these swing options, based on the Fourier cosine expansion method, which is especially suitable when the underlying is modeled by a Levy process.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114559822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-06-01DOI: 10.3970/CMES.2009.047.097
T. B. Jnsthvel, M. B. Gijzen, C. Vuik, C. Kasbergen, A. Scarpas
The introduction of computed x-ray tomography allows for the construction of high quality, material-per-element based 3D meshes in the field of structural mechanics. The use of these meshes enables a shift from meso to micro scale analysis of composite materials like cement concrete, rocks and asphalt concrete. Unfortunately, because of the extremely long execution time, memory and storage space demands, the majority of commercially available finite element packages are not capable of handling efficiently the most computationally demanding operation of the finite element solution process, that is, the inversion of the structural stiffness matrix. To address this issue, an efficient iterative method based upon the preconditioned conjugate gradient method has been developed and is presented in this contribution. It is shown that enhancement of the preconditioned conjugate gradient method with information about the rigid body modes of the aggregates results in an aggregate independent convergence behavior. The resulting number of iterations is bounded by the material behavior of the matrix only.
{"title":"Preconditioned conjugate gradient method enhanced by deflation of rigid body modes applied to composite materials.","authors":"T. B. Jnsthvel, M. B. Gijzen, C. Vuik, C. Kasbergen, A. Scarpas","doi":"10.3970/CMES.2009.047.097","DOIUrl":"https://doi.org/10.3970/CMES.2009.047.097","url":null,"abstract":"The introduction of computed x-ray tomography allows for the construction of high quality, material-per-element based 3D meshes in the field of structural mechanics. The use of these meshes enables a shift from meso to micro scale analysis of composite materials like cement concrete, rocks and asphalt concrete. Unfortunately, because of the extremely long execution time, memory and storage space demands, the majority of commercially available finite element packages are not capable of handling efficiently the most computationally demanding operation of the finite element solution process, that is, the inversion of the structural stiffness matrix. To address this issue, an efficient iterative method based upon the preconditioned conjugate gradient method has been developed and is presented in this contribution. It is shown that enhancement of the preconditioned conjugate gradient method with information about the rigid body modes of the aggregates results in an aggregate independent convergence behavior. The resulting number of iterations is bounded by the material behavior of the matrix only.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123133249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present and evaluate a Fourier-based sparse grid method for pricing multi-asset options. This involves computing multidimensional integrals efficiently and we do it by the Fast Fourier Transform. We also propose and evaluate ways to deal with the curse of dimensionality by means of parallel partitioning of the Fourier transform and by incorporating a parallel sparse grids method. Finally, we test the presented method by solving pricing equations for options dependent on up to seven underlying assets.
{"title":"Multi-asset option pricing using a parallel Fourier-based technique","authors":"C. Leentvaar, C. Oosterlee","doi":"10.21314/JCF.2008.184","DOIUrl":"https://doi.org/10.21314/JCF.2008.184","url":null,"abstract":"In this paper we present and evaluate a Fourier-based sparse grid method for pricing multi-asset options. This involves computing multidimensional integrals efficiently and we do it by the Fast Fourier Transform. We also propose and evaluate ways to deal with the curse of dimensionality by means of parallel partitioning of the Fourier transform and by incorporating a parallel sparse grids method. Finally, we test the presented method by solving pricing equations for options dependent on up to seven underlying assets.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2008-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129529070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compare various numerical methods for the estimation of the VaR and the marginal VaR contribution (VaRC) in the Vasicek one-factor portfolio credit loss model. The methods we investigate are the normal approximation, the saddlepoint approximation, a simplified saddlepoint approximation and importance sampling. We investigate each method in terms of speed, accuracy and robustness and in particular explore their abilities of dealing with exposure concentration.
{"title":"Computation of VaR and VaR Contribution in the Vasicek portfolio credit loss model: A comparative study","authors":"Xinzheng Huang, C. Oosterlee, Mace Mesters","doi":"10.21314/jcr.2007.048","DOIUrl":"https://doi.org/10.21314/jcr.2007.048","url":null,"abstract":"We compare various numerical methods for the estimation of the VaR and the marginal VaR contribution (VaRC) in the Vasicek one-factor portfolio credit loss model. The methods we investigate are the normal approximation, the saddlepoint approximation, a simplified saddlepoint approximation and importance sampling. We investigate each method in terms of speed, accuracy and robustness and in particular explore their abilities of dealing with exposure concentration.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126331176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-12-22DOI: 10.1016/J.JSV.2003.10.049
Zarubinskaya, W. T. Horssen
{"title":"On the free vibrations of a rectangular plate with two opposite sides simply supported and the other sides attached to linear springs","authors":"Zarubinskaya, W. T. Horssen","doi":"10.1016/J.JSV.2003.10.049","DOIUrl":"https://doi.org/10.1016/J.JSV.2003.10.049","url":null,"abstract":"","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124643303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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